In the local case (i.e., \(s_1=s_2= 1\) ), we have Hölder continuity results for the gradient of the weak solutions. Precisely, Lieberman in [1]}, proved \(C^{1,\alpha }(\overline{\Omega })\)  regularity results for quasilinear problems containing more general operators than the classical \(p\) -Laplacian. Subsequently, in [2]}, the author established interior Hölder continuity of the gradient of weak solutions to quasilinear equations whose leading operator has Orlicz type of growth condition. See [3]} for more details on the regularity theory of quasilinear problems.
